A finiteness property of postcritically finite unicritical polynomials

Let $k$ be a number field with algebraic closure $\overline{k}$, and let $S$ be a finite set of places of $k$ containing all the archimedean ones. Fix $d \geq 2$ and $\alpha \in \overline{k}$ such that the map $z \mapsto z^d + \alpha$ is not postcritically finite. Assuming a technical hypothesis on $\alpha$, we prove that there are only finitely many parameters $c \in \overline{k}$ for which $z \mapsto z^d + c$ is postcritically finite and for which $c$ is $S$-integral relative to $(\alpha)$. That is, in the moduli space of unicritical polynomials of degree $d$, there are only finitely many PCF $\overline{k}$-rational points that are $((\alpha), S)$-integral. We conjecture that the same statement is true without the technical hypothesis.

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In memory of Lucien Szpiro

Received 30 October 2020

Received revised 13 May 2023

Accepted 23 June 2023

Published 13 September 2023